A package for representing various bases constructed from basis functions with compact support as quasi-arrays.

This package implements bases with compactly supported bases functions as quasi-arrays where one one axes is continuous and the other axis is discrete (countably infinite), as implemented in QuasiArrays.jl and ContinuumArrays.jl. The bases available initially are various finite-differences (FD), finite-elements discrete-variable representation (FE-DVR), and B-splines, with the possibility of adding more in the future.

Note that this package is written by a pragmatic physicist, so you may find it lacking in mathematical rigour.

The big advantage of this framework is that the user code does not need to be aware of the underlying details of the basis employed, at least that is the goal. As an example, we look at how to construct the mass matrices and second derivative matrices for a few different bases.

```
using CompactBases
function test_basis(B)
println(repeat("-", 100))
display(B)
@info "Mass matrix"
S = B'B
display(S)
# This is the continuous axis
x = axes(B,1)
# This corresponds to a operator L whose action on a function
# f(x) is defined as Lf(x) = sin(2πx)*f(x). In physics this is a
# potential.
@info "Sine operator"
f = QuasiDiagonal(sin.(2π*x))
display(B'*f*B)
@info "Laplacian"
D = Derivative(x)
display(B'*D'*D*B)
println(repeat("-", 100))
println()
end
a,b = 0,1 # Extents
N = 3 # Number of nodes
k = 5 # Order of FE-DVR/B-splines
```

The available finite-differences (as of present) are three-point
stencils, with the first grid point at either `Δx`

(normal) or `Δx/2`

(staggered).

```
Δx = (b-a)/(N+1) # Grid spacing
# Standard, uniform finite-differences
test_basis(FiniteDifferences(N, Δx))
```

```
----------------------------------------------------------------------------------------------------
Finite differences basis {Float64} on 0.0..1.0 with 3 points spaced by Δx = 0.25
[ Info: Mass matrix
UniformScaling{Float64}
0.25*I
[ Info: Sine operator
3×3 Diagonal{Float64,Array{Float64,1}}:
1.0 ⋅ ⋅
⋅ 1.22465e-16 ⋅
⋅ ⋅ -1.0
[ Info: Laplacian
3×3 SymTridiagonal{Float64,Array{Float64,1}}:
-32.0 16.0 ⋅
16.0 -32.0 16.0
⋅ 16.0 -32.0
----------------------------------------------------------------------------------------------------
```

```
# Staggered, uniform finite-differences
test_basis(StaggeredFiniteDifferences(N, Δx))
```

```
----------------------------------------------------------------------------------------------------
Staggered finite differences basis {Float64} on 0.0..0.875 with 3 points spaced by ρ = 0.25
[ Info: Mass matrix
UniformScaling{Float64}
0.25*I
[ Info: Sine operator
3×3 Diagonal{Float64,Array{Float64,1}}:
0.707107 ⋅ ⋅
⋅ 0.707107 ⋅
⋅ ⋅ -0.707107
[ Info: Laplacian
3×3 SymTridiagonal{Float64,Array{Float64,1}}:
-65.25 21.3333 ⋅
21.3333 -35.5556 17.0667
⋅ 17.0667 -33.28
----------------------------------------------------------------------------------------------------
```

The FE-DVR implementation follows

- Rescigno, T. N., & McCurdy, C. W. (2000). Numerical Grid Methods for Quantum-Mechanical Scattering Problems. Physical Review A, 62(3), 032706. http://dx.doi.org/10.1103/physreva.62.032706

The scalar operators are diagonal, whereas differential operators are almost block-diagonal, with one-element overlaps.

```
# Finite-element boundaries
tf = range(a, stop=b, length=N+2)
# By indexing the second dimension, we can implement Dirichlet0
# boundary conditions.
test_basis(FEDVR(tf, max(2,k))[:,2:end-1])
```

```
----------------------------------------------------------------------------------------------------
FEDVR{Float64} basis with 4 elements on 0.0..1.0, restricted to elements 1:4, basis functions 2..16 ⊂ 1..17
[ Info: Mass matrix
UniformScaling{Bool}
true*I
[ Info: Sine operator
15×15 Diagonal{Float64,Array{Float64,1}}:
0.267921 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ 0.707107 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ 0.963441 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ 1.0 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ 0.963441 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ 0.707107 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 0.267921 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 1.22465e-16 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ -0.267921 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ -0.707107 ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ -0.963441 ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ -1.0 ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ -0.963441 ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ -0.707107 ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ -0.267921
[ Info: Laplacian
7×7-blocked 15×15 BlockBandedMatrices.BlockSkylineMatrix{Float64,Array{Float64,1},BlockBandedMatrices.BlockSkylineSizes{Tuple{BlockArrays.BlockedUnitRange{Array{Int64,1}},BlockArrays.BlockedUnitRange{Array{Int64,1}}},Array{Int64,1},Array{Int64,1},BandedMatrix{Int64,Array{Int64,2},Base.OneTo{Int64}},Array{Int64,1}}}:
-746.667 298.667 -74.6667 │ 33.0583 │ ⋅ ⋅ ⋅ │ ⋅ │ ⋅ ⋅ ⋅ │ ⋅ │ ⋅ ⋅ ⋅
298.667 -426.667 298.667 │ -90.5097 │ ⋅ ⋅ ⋅ │ ⋅ │ ⋅ ⋅ ⋅ │ ⋅ │ ⋅ ⋅ ⋅
-74.6667 298.667 -746.667 │ 758.901 │ ⋅ ⋅ ⋅ │ ⋅ │ ⋅ ⋅ ⋅ │ ⋅ │ ⋅ ⋅ ⋅
─────────────────────────────────┼──────────────┼───────────────────────────────────┼──────────────┼───────────────────────────────────┼──────────────┼─────────────────────────────────
33.0583 -90.5097 758.901 │ -2240.0 │ 758.901 -90.5097 33.0583 │ -16.0 │ ⋅ ⋅ ⋅ │ ⋅ │ ⋅ ⋅ ⋅
─────────────────────────────────┼──────────────┼───────────────────────────────────┼──────────────┼───────────────────────────────────┼──────────────┼─────────────────────────────────
⋅ ⋅ ⋅ │ 758.901 │ -746.667 298.667 -74.6667 │ 33.0583 │ ⋅ ⋅ ⋅ │ ⋅ │ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ │ -90.5097 │ 298.667 -426.667 298.667 │ -90.5097 │ ⋅ ⋅ ⋅ │ ⋅ │ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ │ 33.0583 │ -74.6667 298.667 -746.667 │ 758.901 │ ⋅ ⋅ ⋅ │ ⋅ │ ⋅ ⋅ ⋅
─────────────────────────────────┼──────────────┼───────────────────────────────────┼──────────────┼───────────────────────────────────┼──────────────┼─────────────────────────────────
⋅ ⋅ ⋅ │ -16.0 │ 33.0583 -90.5097 758.901 │ -2240.0 │ 758.901 -90.5097 33.0583 │ -16.0 │ ⋅ ⋅ ⋅
─────────────────────────────────┼──────────────┼───────────────────────────────────┼──────────────┼───────────────────────────────────┼──────────────┼─────────────────────────────────
⋅ ⋅ ⋅ │ ⋅ │ ⋅ ⋅ ⋅ │ 758.901 │ -746.667 298.667 -74.6667 │ 33.0583 │ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ │ ⋅ │ ⋅ ⋅ ⋅ │ -90.5097 │ 298.667 -426.667 298.667 │ -90.5097 │ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ │ ⋅ │ ⋅ ⋅ ⋅ │ 33.0583 │ -74.6667 298.667 -746.667 │ 758.901 │ ⋅ ⋅ ⋅
─────────────────────────────────┼──────────────┼───────────────────────────────────┼──────────────┼───────────────────────────────────┼──────────────┼─────────────────────────────────
⋅ ⋅ ⋅ │ ⋅ │ ⋅ ⋅ ⋅ │ -16.0 │ 33.0583 -90.5097 758.901 │ -2240.0 │ 758.901 -90.5097 33.0583
─────────────────────────────────┼──────────────┼───────────────────────────────────┼──────────────┼───────────────────────────────────┼──────────────┼─────────────────────────────────
⋅ ⋅ ⋅ │ ⋅ │ ⋅ ⋅ ⋅ │ ⋅ │ ⋅ ⋅ ⋅ │ 758.901 │ -746.667 298.667 -74.6667
⋅ ⋅ ⋅ │ ⋅ │ ⋅ ⋅ ⋅ │ ⋅ │ ⋅ ⋅ ⋅ │ -90.5097 │ 298.667 -426.667 298.667
⋅ ⋅ ⋅ │ ⋅ │ ⋅ ⋅ ⋅ │ ⋅ │ ⋅ ⋅ ⋅ │ 33.0583 │ -74.6667 298.667 -746.667
----------------------------------------------------------------------------------------------------
```

All operators become banded when using B-splines, including the mass matrix, which leads to generalized eigenvalue problems, among other things.

```
tb = LinearKnotSet(k, a, b, N+1)
test_basis(BSpline(tb)[:,2:end-1])
```

```
----------------------------------------------------------------------------------------------------
BSpline{Float64} basis with LinearKnotSet(Float64) of order k = 5 (quartic) on 0.0..1.0 (4 intervals), restricted to basis functions 2..7 ⊂ 1..8
[ Info: Mass matrix
6×6 BandedMatrix{Float64,Array{Float64,2},Base.OneTo{Int64}}:
0.0414683 0.0307567 0.00934744 0.00101411 2.75573e-6 ⋅
0.0307567 0.0581184 0.0437077 0.0124663 0.000610548 2.75573e-6
0.00934744 0.0437077 0.0786449 0.0543455 0.0124663 0.00101411
0.00101411 0.0124663 0.0543455 0.0786449 0.0437077 0.00934744
2.75573e-6 0.000610548 0.0124663 0.0437077 0.0581184 0.0307567
⋅ 2.75573e-6 0.00101411 0.00934744 0.0307567 0.0414683
[ Info: Sine operator
6×6 BandedMatrix{Float64,Array{Float64,2},Base.OneTo{Int64}}:
0.0243786 0.0237789 0.00817377 0.000914686 1.89458e-6 ⋅
0.0237789 0.0508203 0.0345991 0.00707979 7.42166e-20 -1.89458e-6
0.00817377 0.0345991 0.034669 5.37956e-18 -0.00707979 -0.000914686
0.000914686 0.00707979 6.40955e-18 -0.034669 -0.0345991 -0.00817377
1.89458e-6 5.7276e-20 -0.00707979 -0.0345991 -0.0508203 -0.0237789
⋅ -1.89458e-6 -0.000914686 -0.00817377 -0.0237789 -0.0243786
[ Info: Laplacian
6×6 BandedMatrix{Float64,Array{Float64,2},Base.OneTo{Int64}}:
-7.08571 -0.530159 1.01587 0.253968 0.0031746 ⋅
-0.530159 -2.53333 -0.26455 0.756966 0.161552 0.0031746
1.01587 -0.26455 -1.8455 -0.310406 0.756966 0.253968
0.253968 0.756966 -0.310406 -1.8455 -0.26455 1.01587
0.0031746 0.161552 0.756966 -0.26455 -2.53333 -0.530159
⋅ 0.0031746 0.253968 1.01587 -0.530159 -7.08571
----------------------------------------------------------------------------------------------------
```